Explanation Given: The function is f ( x ) = − 4 x 1 + 2 x . Definition used: (1) If the function gets larger and larger in magnitude without bound as x approaches the number k, then the line x = k is the vertical asymptote. (2) A function f ( x ) said to be symmetric about the origin if f ( − x ) = − f ( x ) . Calculation: Domain: The given function f ( x ) = − 4 x 1 + 2 x is defined at all the real number except the point x = − 1 2 . Thus, the domain of the function f ( x ) = − 4 x 1 + 2 x is ( − ∞ , − 1 2 ) ∪ ( − 1 2 , ∞ ) . Symmetry: The given function is f ( x ) = − 4 x 1 + 2 x . Substitute x = − x in f ( x ) , f ( − x ) = − 4 ( − x ) 1 + 2 ( − x ) = 4 x 1 − 2 x Thus, f ( − x ) ≠ − f ( x ) . Therefore, from the definition stated above, the function does not have symmetry about the origin. Intercept: The function f ( x ) intercepts y -axis at x = 0 . Substitute the value x = 0 in f ( x ) , f ( 0 ) = − 4 ( 0 ) 1 + 2 ( 0 ) = 0 That is the function f ( x ) intercepts y -axis at the point ( 0 , 0 ) . The function f ( x ) intercepts x -axis at f ( x ) = 0 . − 4 x 1 + 2 x = 0 − 4 x = 0 x = 0 Thus, x = 0 . That is the function f ( x ) intercepts x -axis at the points ( 0 , 0 ) . Asymptote: Observe that the graph of the function f ( x ) = − 4 x 1 + 2 x tends to the line x = − 1 2 as the value of x tends to − 1 2 . Thus x = − 1 2 is the vertical asymptote for the function f ( x ) = − 4 x 1 + 2 x . Observe the limit, lim x → ∞ f ( x ) = lim x → ∞ − 4 x 1 + 2 x = lim x → ∞ − 4 1 x + 2 = − 4 2 = − 2 . Thus, as the value of x tends to infinity, the graph of the function f ( x ) = − 4 x 1 + 2 x tends to the line y = − 2 . Thus y = − 2 is the horizontal asymptote for the function f ( x ) = − 4 x 1 + 2 x . Critical point: The given function is f ( x ) = − 4 x 1 + 2 x . Differentiate f ( x ) with respect to x , f ′ ( x ) = − 4 ( 1 + 2 x ) − 2 ( − 4 x ) ( 1 + 2 x ) 2 = − 4 − 8 x + 8 x ( 1 + 2 x ) 2 = − 4 ( 1 + 2 x ) 2 Observe that the function f ′ ( x ) = − 4 ( 1 + 2 x ) 2 is never zero

Calculus with Applications (11th Edition)

11th Edition

ISBN: 9780321979421

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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